CN115014637A - Modal dynamic balance method based on low-rotation-speed measurement - Google Patents

Modal dynamic balance method based on low-rotation-speed measurement Download PDF

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CN115014637A
CN115014637A CN202210411949.5A CN202210411949A CN115014637A CN 115014637 A CN115014637 A CN 115014637A CN 202210411949 A CN202210411949 A CN 202210411949A CN 115014637 A CN115014637 A CN 115014637A
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邓振鸿
罗华耿
张保强
吴太欢
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Xiamen University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M1/00—Testing static or dynamic balance of machines or structures
    • G01M1/14—Determining imbalance
    • G01M1/16—Determining imbalance by oscillating or rotating the body to be tested
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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    • Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation

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Abstract

A modal dynamic balance method based on low rotation speed measurement relates to the field of rotating machinery and vibration testing. In order to eliminate the unbalanced response of the previous N-order mode, adding test weights on N planes, solving the unbalanced calibration quantity required to be added, taking N balance surfaces to balance the previous N-order mode, obtaining the synchronous vibration response of a measuring point when the rotor operates at any rotating speed, obtaining the previous N + M-order mode at a lower rotating speed, measuring different rotating speeds within a range lower than the first-order critical rotating speed of the rotor, adding a known test weight on the first balance surface, obtaining an influence coefficient after measuring different rotating speeds in the same way, and repeatedly solving the unbalanced calibration quantity on the remaining balance surfaces. The rotor can be measured at a lower rotating speed, and the balance parameters are fitted by combining the rotor modal parameter information, so that modal dynamic balance at a low speed is realized, resonance caused by passing a critical rotating speed in the test process is avoided, and a more reliable balance state of the rotor in a wider rotating speed domain is ensured.

Description

Modal dynamic balance method based on low-rotation-speed measurement
Technical Field
The invention relates to the field of rotating machinery and vibration testing, in particular to a modal dynamic balance method based on low-rotation-speed measurement, which can be applied to the dynamic balance of a flexible rotor.
Background
Industrial rotary machines such as aircraft engines and large gas turbines have high operating speeds, rotors of the industrial rotary machines often need to work in a supercritical speed state, and the rotors often have significant vibration and deformation due to structural resonance in the process of passing through the critical speed due to unbalanced rotor mass, so that the operating performance and safety of equipment are seriously affected.
In order to ensure that a rotor system can pass through the critical rotating speed more stably, the conventional modal dynamic balance method generally requires dynamic balance test near the critical rotating speed, but the problem of overlarge vibration when the rotor system is close to or passes through the critical rotating speed exists in the measuring process, so that the dynamic balance test cost is higher, and greater potential safety hazards exist. Therefore, a new dynamic balancing method which is safer and more reliable is urgently needed.
Disclosure of Invention
The invention aims to provide a modal dynamic balance method based on low rotating speed measurement, so that a rotor can be measured at a lower rotating speed, resonance is avoided as far as possible in the test process, modal balance is realized, and a more reliable balance state of the rotor in a wider rotating speed domain is ensured.
The method comprises the following specific steps:
1) for flexible rotor systems, it is in the axial direction x s The synchronous vibration response at the points due to mass imbalance is as follows:
Figure BDA0003604084170000011
wherein ,mr ，ω r ，ξ r ，ψ r (x) Respectively representing the r-order modal mass, modal frequency, modal damping and modal shape, wherein omega is the shaft rotating speed, U (x) is the unbalanced distribution of the rotor, and l is the length of the rotor; generally, when the rotation speed approaches the modal frequency of the r-th order, the modal response of this order dominates, written as:
Figure BDA0003604084170000021
to eliminate the unbalanced response of the former N-th order mode, it is necessary to add trial weights λ on N planes k N, such that k is 1,2
Figure BDA0003604084170000022
Comprises the following steps:
Figure BDA0003604084170000023
in the formula ,
Figure BDA0003604084170000024
representing the integral of the nth order modal imbalance component over the entire axial direction, which may be referred to as the nth order modal imbalance factor, equation (3) is written in the form of a matrix:
Figure BDA0003604084170000025
the amount of imbalance calibration that needs to be added can be solved from the above equation:
Λ＝-[Ψ] -1 u (5)
wherein Λ ═ λ 1 ,λ 2 ,...λ N ] T Representing the set of imbalance calibration quantities that need to be added, [ psi [ [ phi ]] r ＝[ψ r (x 1 ),ψ r (x 2 ),...,ψ r (x N )]Represents the r-th order mode vector, u ═ u 1 ,u 2 ,...u N ] T And represents a vector consisting of the modal imbalance factors of the first N orders.
2) Taking N balance surfaces to balance the former N-order mode, and setting the measuring point at x s Where each order of mode shape is psi i (x s ) Dividing each row [ psi ] of the mode matrix] i Ride on
Figure BDA0003604084170000026
Then the write is:
Figure BDA0003604084170000027
when the rotor runs at any rotating speed, the rotor is at a measuring point x s The synchronous vibration response of (a) is:
Figure BDA0003604084170000028
wherein ,
Figure BDA0003604084170000029
defined as a complex parameter term related to the rotation speed, when the mode of the first N orders is considered to be balanced, the mode after the N + M orders is considered to have negligible contribution to the first N orders, and then the rotation speed omega is lower (omega < omega) 1 ＜ω N ＜＜ω N+M+1 ) Next, truncating the above formula to the first N + M order mode to obtain:
Figure BDA0003604084170000031
when different rotation speeds omega are carried out in a range lower than the first-stage critical rotation speed of the rotor, { omega ═ omega 1 ,Ω 2 ,...Ω p The measurement of (p.gtoreq.N + M) is then written as:
Figure BDA0003604084170000032
for simplicity, remember
Figure BDA0003604084170000033
Z＝[Z 1 (Ω) Z 2 (Ω)...Z N+M (Ω)]When the critical rotation speed ω of each stage of the rotor is known i (i ═ 1, 2.., n + m) and the critical damping ratio ξ i (i ═ 1, 2., n + m), then solved by the above equation:
Figure BDA0003604084170000034
in the formula ,(·)+ Representing the pseudo-inverse of the matrix, Y ═ Y (x) s ,Ω 1 ),Y(x s ,Ω 2 ),...,Y(x s ,Ω p )] T 。
Adding a known trial weight Q to the first balance surface 1 Similarly, different rotation speed Ω ═ Ω is performed 1 ,Ω 2 ,...Ω p The influence coefficient is obtained after the measurement of (p is more than or equal to N + M):
Figure BDA0003604084170000035
writing the above equation in matrix form:
H 1 ＝ZX 1 (12)
wherein ,
Figure BDA0003604084170000036
represents a term related to mode shape and modal mass, so:
Figure BDA0003604084170000041
repeating the steps for the rest balance surfaces to obtain:
Figure BDA0003604084170000042
wherein ,
Figure BDA0003604084170000043
and solving the unbalance calibration quantity lambda.
Compared with the prior art, the invention has the beneficial effects that:
the premise of modal balance is that the modal shape is known, and the rotor needs to be operated to be close to the critical rotating speed of each order for measurement, but is limited by practical conditions, such as the rotor structure is complex, the shape of the vibration cannot be accurately obtained, and the operation of the rotor to the critical rotating speed is dangerous and difficult to realize, so the invention adopts a low-speed modal balance step, the rotor can be measured under the condition of being lower than the critical rotating speed, the modal balance can be realized, the balance process only needs 1 sensor, and the modal shape of the rotor does not need to be known in advance.
Drawings
FIG. 1 is a schematic view of a rotor imbalance distribution.
Fig. 2 is a plot of the first 5 th order mode shape of the rotor.
FIG. 3 is a vibration response curve of a rotor due to imbalance.
FIG. 4 is a measurement of a vibration response curve under initial imbalance.
FIG. 5 is a comparison of vibration response amplitude before and after balancing.
Detailed Description
The invention is further illustrated by reference to the following examples.
The embodiment adopted by the invention is a numerical simulation case, which is specifically described as follows: fig. 1 shows a uniform cross-section rotor with a length of 1m, and unbalanced masses are present at different cross-section positions, the specific positions and sizes of which are shown in table 1. The maximum working rotating speed of the rotor is set to be between the third-order critical rotating speed and the fourth-order critical rotating speed, and the contribution of the sixth-order mode and the modes above to the vibration response of the rotor in the whole working rotating speed is considered to be negligible, so that the real vibration response of the rotor is represented by the superposition result of the first 5-order mode components, the first 5-order critical rotating speed, the damping ratio and the mode mass of the rotor are set to be shown in table 2, and the first 5-order mode vibration type curve of the rotor is shown in fig. 2. The imbalance response curve of the rotor in the first 5 critical speed range can be calculated from the given parameters as shown in fig. 3.
TABLE 1 magnitude and distribution of unbalance
Figure BDA0003604084170000051
TABLE 2 critical speed to damping ratio of first 5 stages of rotor
Figure BDA0003604084170000052
In this embodiment, the target is to balance the first 3-order modal imbalance response, and the first 5-order critical rotation speed and damping ratio information of the rotor are estimated through simulation or experiment, and the remaining information (the imbalance distribution, the modal mass, and the modal shape are all unknown quantities) of the rotor is implemented as follows:
1) selecting U 1 、U 2 and U3 The plane is used as a balance plane, and the position of the sensor is arranged in a U 2 The plane is located;
2) response Y to rotor imbalance in the low speed range (200- 0 (Ω,
Figure BDA0003604084170000056
) The amplitude and phase of the vibration are measured, and an unbalanced vibration response curve is obtained and is shown in fig. 4;
3) the information of the critical rotating speed and the damping ratio of the first 5 th order of the rotor is estimated through simulation or experiment (see table 2), and the formula is utilized
Figure BDA0003604084170000053
Z is calculated within the measuring rotating speed range omega of 200-600RPM r (Ω), r is 1,2,3,4,5, further solving:
Figure BDA0003604084170000054
4) at U 1 Adding a known test weight block Q at the position of the plane 1 (magnitude of 1 g.mm, phase of 0 °), and the same measurement was carried out at 200-
Figure BDA0003604084170000055
Calculate H 1 (Ω), and solving:
Figure BDA0003604084170000061
5) take off Q 1 In U 2 Adding a known test weight block Q at the position of the plane 2 (magnitude of 1 g.mm, phase of 0 °), again measured at 200-
Figure BDA0003604084170000062
Calculate H 2 (Ω), solving:
Figure BDA0003604084170000063
6) take off Q 2 In U at 2 Adding a known test weight block Q at the position of the plane 3 (magnitude of 1 g.mm, phase of 0 °), again measured at 200-
Figure BDA0003604084170000064
Calculate H 3 (Ω), solve:
Figure BDA0003604084170000065
7) gamma and X obtained in the steps 3) to 6) 1 、X 2 and X3 In the formula, the balance calibration amounts required to be added on the three balance surfaces are obtained as shown in table 3:
TABLE 3
Calibration quantity λ 1 λ 2 λ 3
Size (g mm) 0.44 1.69 0.36
Phase (°) 14.961 -89.86 -27.28
8) And (4) balancing according to the balance calibration quantity obtained in the step (7), and finishing the balance of the rotor. For example, as shown in fig. 5, the imbalance response curve (dashed line in the figure) after the balancing according to the method of the present invention is significantly reduced in the entire operating rotational speed range compared to the imbalance response curve (solid line in the figure) before the balancing, and the vibration response caused by the modal imbalance in the rotational speed range can be greatly eliminated.

Claims (1)

1. A modal dynamic balance method based on low rotation speed measurement is characterized by comprising the following specific steps:
1) for flexible rotor systems, it is in the axial direction x s The synchronous vibration response at the points due to mass imbalance is as follows:
Figure FDA0003604084160000011
wherein ,mr ，ω r ，ξ r ，ψ r (x) Respectively representing the r-order modal mass, modal frequency, modal damping and modal shape, wherein omega is the shaft rotating speed, U (x) is the unbalanced distribution of the rotor, and l is the length of the rotor; generally, when the speed of rotation approaches the modal frequency of the r-th order, the modal response of that order dominates, written as:
Figure FDA0003604084160000012
to eliminate the unbalanced response of the former N-th order mode, it is necessary to add trial weights λ on N planes k N, such that k is 1,2
Figure FDA0003604084160000013
Comprises the following steps:
Figure FDA0003604084160000014
in the formula ,
Figure FDA0003604084160000015
written in matrix form:
Figure FDA0003604084160000016
the amount of imbalance calibration that needs to be added can be solved according to the above equation:
Λ＝-[Ψ] -1 u
wherein Λ ═ λ 1 ,λ 2 ,…λ N ] T ，[Ψ] r ＝[ψ r (x 1 ),ψ r (x 2 ),…,ψ r (x N )]，u＝[u 1 ,u 2 ,…u N ] T ；
2) Taking N balance surfaces to balance the former N-order mode, and setting the measuring point at x s Where each order of mode shape is psi i (x s ) Dividing each row [ psi ] of the mode matrix] i Ride on
Figure FDA0003604084160000017
Then the write is:
Figure FDA0003604084160000021
when the rotor runs at any rotating speed, the rotor is at a measuring point x s The synchronous vibration response of (a) is:
Figure FDA0003604084160000022
wherein ,
Figure FDA0003604084160000023
when considering the balanced first N-order modes, the mode after the N + M order is considered to have negligible contribution to the first N order, and then at a lower rotation speed omega (omega < omega) 1 ＜ω N ＜＜ω N+M+1 ) Next, truncating the above formula to the first N + M order mode to obtain:
Figure FDA0003604084160000024
when different rotation speeds omega are carried out in a range lower than the first-stage critical rotation speed of the rotor, { omega ═ omega 1 ,Ω 2 ,…Ω p The measurement of (p.gtoreq.N + M), the above equation is written as:
Figure FDA0003604084160000025
note the book
Figure FDA0003604084160000026
Z＝[Z 1 (Ω) Z 2 (Ω)...Z N+M (Ω)]When the critical rotation speed ω of each stage of the rotor is known i (i ═ 1, 2.., n + m) and the critical damping ratio ξ i (i ═ 1,2,. ang., n + m), then solved by the above equation:
Figure FDA0003604084160000027
in the formula ,(·)+ Representing the pseudo-inverse of the matrix, Y [ [ Y (x) ] s ,Ω 1 ),Y(x s ,Ω 2 ),...,Y(x s ,Ω p )] T ；
Adding a known trial weight Q to the first balance surface 1 Similarly, different rotation speed Ω ═ Ω is performed 1 ,Ω 2 ,...Ω p The influence coefficient is obtained after the measurement of (p is more than or equal to N + M):
Figure FDA0003604084160000031
writing the above equation in matrix form:
H 1 ＝ZX 1
wherein ,
Figure FDA0003604084160000032
therefore, the method comprises the following steps:
Figure FDA0003604084160000033
repeating the steps for the rest balance surfaces to obtain:
Figure FDA0003604084160000034
wherein ,
Figure FDA0003604084160000035
and solving the unbalance calibration quantity lambda.
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